Abstract Quad-morphing is a new technique used for generating quadrilaterals from an existing triangle mesh. Beginning with an initial triangulation, triangles are systematically transformed and combined. An advancing front method is used to determine the order of transformations. An all-quadrilateral mesh containing elements aligned with the area boundaries with few irregular internal nodes can be generated. KEY WORDS: mesh generation, quadrilateral, advancing front, surface meshing, Q-Morph, Paving

1. Introduction Previous methods for unstructured quadrilateral meshing have included both direct and indirect methods. Indirect methods (Lo,1989; Johnston,1991; Lee,1994; Borouchaki,1998) include procedures that require an initial triangle mesh. Adjacent triangles are combined systematically, in most cases resulting in an allquadrilateral mesh. While these methods can be fast, they can sometimes leave a large number of irregular nodes. An irregular node on the interior of a quadrilateral mesh is one that has more or less than four adjacent elements. Direct methods, on the other hand, do not involve an initial triangle mesh. Quadrilaterals are instead placed directly onto the surface. Quadrilaterals may be placed after first decomposing the surface into simpler regions (Baehmann,1987; Talbert,1991; Tam,1991; Joe,1995) or by using an advancing front approach (Zhu,1991; Lo,1985; Blacker,1991). In most cases, direct methods provide higher quality elements with fewer irregular nodes. Of the direct, quadrilateral methods, the paving algorithm (Blacker,1991) provides several desirable characteristics. Blacker describes these as “(a) Boundary Sensitive. Mesh contours should closely follow the contours of the boundary. This characteristic is of particular importance since well-shaped elements are usually desirable near the boundary, (b) Orientation Insensitive. Rotating or translating a given geometry should not change the resulting mesh topology. A mesh generated in a transformed geometry should be equivalent to the original mesh transformed, and (c) Few irregular nodes. This is a critical mesh topology feature because the number of elements sharing a node controls the final shape of the elements, even after smoothing. Thus a mesh with few irregular nodes, especially near the boundary where element shape is critical, is often preferred.”. The paving algorithm is currently in wide use. Since its initial development, it has been enhanced to incorporate three-dimensional surfaces (Cass,1996), as well as other improvements (White,1997).

In spite of the beneficial characteristics of paving, some quality and performance issues must be addressed. The advancing front method used by the paving technique requires many expensive intersection calculations as each row is placed in order to avoid overlapping elements. Figure 1(a) shows a simple case where intersection checks must be made. Figure 1(b) shows another case often encountered during paving where colliding fronts must merge. If element sizes differ greatly, poor element quality can often result.

(a)

(b)

Figure 1. (a) First row of elements placed using paving algorithm illustrating interference of opposing elements. (b) Large element size differences between opposing fronts often encountered in paving leading to poor meshes.

This paper proposes an alternative to the traditional paving algorithm. The proposed Quad-morphing (QMorph) algorithm maintains the desirable features of paving while addressing some of its weaknesses. QMorph can be categorized as an unstructured, indirect method that utilizes an advancing front algorithm to form an all-quad mesh. As an indirect method it is able to take advantage of local topology information from the initial triangulation. Unlike other indirect methods it is able to generate boundary sensitive rows of elements, with few irregular nodes.

2. Outline of Quad-Morphing Algorithm Quad-morphing is briefly outlined in the following steps: 1. Initial Triangle Mesh. The surface is first triangulated. This may be done using any surface triangulation method. Any sizing (Owen,1997) or adaptivity information should be built into the initial triangulation. The local sizing for the final quadrilateral mesh will roughly follow that of the triangle mesh. Front Definition. The initial front is defined from the initial triangle mesh. Any edge in the triangulation that is adjacent to only one triangle becomes part of the initial front. Front Edge Classification. Each edge in the front is initially sorted according to its state. The state of a front edge defines how the edge will eventually be used in forming a quadrilateral. Angles between adjacent front edges determine the state of an individual front. Front edges will be updated and reshuffled as the algorithm proceeds. Figure 2 shows the four possible states of a front, where the front edge is indicated by the bold line.

2. 3.

the local triangulation is modified by using local edge swaps to
•
•
. side edge NB-NC shows the use of an existing edge. by swapping the diagonal of adjacent triangles. Side edges may be defined by using an existing edge in the initial triangle mesh. Using the front edge as the initial base edge of the quadrilateral. Top Edge Recovery. or by splitting triangles to create a new edge. This process can be further subdivided into the following substeps:
ND NC NA NB (a) Initial front NA NB (b) Side edge definition
ND NC NA NB (c) Top edge recovery
ND NC NA NB (d) Quadrilateral formation
ND
NC
NA NB (e) Local smooth
Figure 3. Front edges are handled differently according to their current state classification.
Front Edge Processing. States of a front edge
4. Figure 3(a) shows front NA-NB in the triangulation ready to be processed. the front is redefined and adjacent front edge states are updated. side edges are defined. while the side edge NA-ND was formed from a local swap operation. As quadrilaterals are formed. Each front edge is individually processed to create a new quadrilateral from the triangles in the initial mesh. In Figure 3(b). Side Edge Definition. During this process. several special case scenarios are checked. or transition seam operation is performed.State 0-0
State 1-0
State 0-1
State 1-1
Figure 2. The current front always defines the interface between quadrilateral elements in the final mesh and triangle elements in the initial triangle mesh. The final edge on the quadrilateral is created by an edge recovery process. Before proceeding to construct a quadrilateral from the current front. These include situations where large transitions or small angles exist local to the front. •
Check for Special Cases. In these cases a seam. Steps demonstrating process of generating a quadrilateral from Front NA-NB.

The mesh is smoothed locally to improve both quadrilateral and triangle element quality as shown in Figure 3(e). the node bit is set (1). followed by edges in state 0-0. otherwise it is unset (0). it defines which edges must be defined before a complete quadrilateral can be formed. When an odd number of boundary intervals is provided. angles must be computed by first evaluating the surface normal and projecting edge vectors to a tangent plane.1996). Local Front Reclassification. the first representing the state at the left node and the second. All edges in the triangulation adjacent to a single triangle are used as the front. 5. Front edges in state 1-1 are given first priority followed by edges in states 0-1 and 1-0. Element quality is improved by performing local quadrilateral transformations in an attempt to improve the individual edge valences at the nodes of the mesh. in which case an allquadrilateral mesh will remain.
3. Practically. assuming an even number of initial front edges. If the angle at either node is less than a specified tolerance (3π/4).
.
Front edge processing continues until all edges on the front have been depleted. The front is advanced by removing edges from the front that have two quadrilateral adjacencies and adding edges to the front that have one triangle and one quadrilateral adjacency. Edge NC-ND in Figure 3(c) was formed from a single swap operation. Side edges must be defined only at the side of the front where the state bit has not been set. a single triangle must be generated. The state of a front edge is determined by computing the angle at the nodes on either end of the edge with each of its adjacent front edges. 6. Topological Clean-up. Existing fronts that may have been adjusted in the smoothing process are reclassified.enforce an edge between the two nodes at the ends of the two side edges.
• • •
Quadrilateral Formation. Merging any triangles bounded by the front edge and the newly created side edges and top edge as shown in Figure 3(d) forms the final quadrilateral . A final smoothing pass is performed further improving the element qualities. Edges are placed on one of four state lists as shown in Figure 2. expensive geometric evaluations can be eliminated. New front edges are classified by state. First.1 Front Definition and Classification The initial set of front edges is defined from the initial triangulation. Implementation
3. By approximating the angle at the front from the adjacent triangles. Any number of swaps may be required to form the top edge. In direct advancing front methods (Cass. Angles at the nodes on the front can be approximated by summing the angles at adjacent triangles. Smoothing. Second. it prioritizes which fronts will be processed first. Local Smoothing. usually towards the interior of the mesh. the state of a front edge is defined by two bits. Classifying front edges according to states serves two purposes. the state at the right node.

The edge with the smallest angle θ is selected as the candidate side edge. one of two options may be used. based not only on the current state and level of the front but also on its size. Edge E2 in Figure 4 is selected as the side edge in this situation. Edges in level zero are those on the initial front. The edge is selected. higher level fronts before selecting longer lower level fronts. (Eo in Figure 5) may either be swapped or split. Ei.3. Side edge selection
Figure 4 shows a situation in which an existing edge is used. Priority is also given to the lowest level edge on the list. to be used as a side edge in the new quadrilateral:
. This ensures that an entire row of quadrilaterals will be placed before starting a new row. experience has shown that placing smaller quads first generally produces a better graded mesh. Edges Ek and Em are also added to the triangle mesh. The criteria used for selecting the next front to be processed is. The following shows a summary of the criteria for selection or creation of edge Ek.1 Side Edge Definition The current state of a front edge determines how the edge is processed. 3. The opposite edge. Front edges in states 0-0. 1-0 and 01 must first define either one or two side edges. (2) the diagonal between two adjacent triangles may be swapped. therefore. level one are those on the front after the first row of quadrilaterals has been placed. which is a node on the front between edges EF1 and EF2. When there is no angle θi less than ε. it is sometimes necessary to select short. splitting the two triangles adjacent to edge Eo. of triangles sharing node Nk. or (3) an edge may be created by splitting a pair of triangles. level two after the second row. splitting edge Eo at the intersection of vector Vk and edge Eo.
Vk
ε θ2 ε θ1
Eo
E2 EF1 Nk
E1 EF2
Figure 4. Edge Ek is then used as the side edge of the prototype quad. A new side edge is to be defined at node Nk.2. a new node Nn is defined. provided θ is less than a constant ε (π/6). Where large transitions are required. The split option is performed if β > ε or the resulting length of Ek from a swap is excessively long compared to EF1 and EF2. A side edge may be formed in one of three ways: (1) an existing edge in the initial triangle mesh may be used. In order to do so. In this latter case. The swap option is used if the angle β between Vk and Vm is less than ε. drawing from the higher states first. Angles θi are computed between Vk and all edges. The ideal vector Vk for the new side edge is defined by bisecting the vectors formed by EF1 and EF2. A front edge is popped from one of the four state lists.2 Front Edge Processing Front edges are processed one at a time to form quadrilaterals from the initial triangulation. and so on.

the next step is to define the top edge. An example of an edge recovery process is shown in Figure 6. Algorithm 1 details the transformations necessary to accomplish the recovery. This is done by recovering the edge between the end nodes of the two sides. which was presented independently in the literature by Jones (1990). In this example a total of four local swaps was required to recover the edge NC-ND from the triangulation.
.2 Top Edge Recovery Once the base and the two sides of the quadrilateral have been formed. and George (1991). Sloan (1993). Side edge creation
3.2.Ei    E k = swap ⇒ N k N m   split ⇒ N k N n 
for θi < ε E F1 + E F2 for β < ε and N k N m < 3 2 otherwise
      
[1]
Nm
(a) swap Vm Nm
β
Ek E2 EF2 E1
ε θ2
Vk
Nk
EF2
ε
Eo
θ1
E2 EF2
E1 Nm Vk
Nk
EF2 Em Nn E2 EF2 (b) split Ek
E1
Nk
EF2
Figure 5. The triangulation before recovery is shown at the top left with the successive swaps numbered. Edge recovery involves systematically swapping edges between adjacent triangles until an edge is achieved between the desired nodes. is used commonly in boundary constrained Delaunay triangle meshing. The edge recovery technique.

FOR EACH Ei ∈ Λ(S) 4. LET T(Ei) be the set of 2 triangles adjacent Ei 5. Form T-1(Ei) 8. Edge Recovery Process
1. LET S be the line segment from NC to ND 2.2. an extension of the edge recovery process to include three-dimensional surfaces was necessary. The dot product calculation in step 16 can then be replaced as:
. 3. Delete Ei from Λ(S) 9. IF area of both triangles in T-1(Ei) > 0 THEN 7. LET Λ(S) be a list of edges Ei that are intersected by S (see algorithm 2) 3. LET Ej be the edge common to both triangles in T-1(Ei) 10. be first compiled. the tangent plane normal Pi at edge Ei can be estimated from the average normal vector of triangles Ti(Ei) and Ti+1(Ei). 6. Edge recovery process
The edge recovery process requires that an initial set of edges. Λ(S). Algorithm 2 and Figure 7 detail how this may be accomplished. For example.3 3D Edge Recovery Current literature assumes that the edge recovery process will be performed on a planar domain.T(Ei) S ND Ei NC ND
T-1(Ei) 1 Ej 2 NC
(a) Initial triangulation
(b) Swap 1. Since this condition cannot be guaranteed in this application. through which the recovered edge will pass. LET T-1(Ei) be the set of 2 triangles where the diagonal edge Ei has been swapped. IF Ej intersects S add Ej last on Λ(S). 11. NEXT Ei on Λ(S)
Algorithm 1. The tangent plane can be approximated from the neighboring triangles. the dot product calculations of steps 5 and 16 in Algorithm 2 must be performed on vectors in a plane that is tangent to the surface. Specifically. Place Ei last on Λ(S) 12. ELSE.2
ND 3
NC
ND
4
NC
(c) Swap 3
(d) Swap 4 Edge Recovered
Figure 6.

Unused nodes and edges are also removed. In practice.1988) is adequate.3. This includes nodes both behind and in front of the current front. As a result. an improved transition can be achieved. This can be done with a procedure that starts with the triangle adjacent to the front edge and recursively advancing to adjacent triangles. For smaller transitions.1976) followed by corrections for squareness and angle smoothness. two side edges. then lD can be defined as an average of edge lengths on adjacent quadrilaterals and edges ahead of the front as follows:
.4 Local Smoothing Smoothing is an important part of the Q-Morph algorithm. Definition of edge length lD at node on front Nk
Let lD be the length of the edge Nk-Nj where Nk is the node on the front to be smoothed as shown in Figure 8. The smoothing process presented by Blacker involves an isoparametric smooth (Hermann. a simple Laplacian smooth (Field. a modified length weighted Laplacian smooth as suggested by Blacker (1991). or alternatively. Node locations local to the new quadrilateral are readjusted to improve element shape. The recursion continues until the top or side edges of the prototype quadrilateral are encountered. A modified form of the smoothing process suggested by Blacker is used for the row nodes defined in that reference. as smoothing angles between adjacent fronts will affect the front states and hence the final topology of the quadrilateral mesh. lD can be defined as the average length of all edges connected to Nk. 3. deleting them as it proceeds. Let n be the number of nodes ahead of the front connected to Nk. Nodes on the front must be handled differently than those behind or ahead of the front. These are nodes connected to exactly two adjacent quadrilaterals at the front. Before forming the quadrilateral. fewer irregular nodes will be created if equation 4 is used. For cases where large transitions may be involved.3 Quadrilateral Formation The quadrilateral is formed from the edge on the front.
Nt2 Nt1
Nk-1 Nk lD Nk+1
Nj-1 Nj Nj+1
Figure 8. Since it is at the front where the new quadrilateral elements are formed. it is more critical at these nodes that the smoothing produce well proportioned quadrilaterals. For nodes not located on the current front. and recovered top edge. Where a very large transition in element size is required. Local smoothing accomplished before processing the next front. any node on the new quadrilateral and any node connected by an edge are smoothed. it is useful to take advantage of sizing information provided by the triangles ahead of the front. the triangles contained within the four edges must first be deleted.

Equation 4 is used for lD when tr is greater than 2. other nearby edges on the front may need to be updated. an adjustment must be made to the new node location. thus necessitating moving fronts to an alternate state list.5 Local Update and Reclassification of Fronts Once a new quadrilateral has been formed. tr. For the current implementation. triangles and quadrilaterals neighboring the smoothed node must be checked for consistent normals. angles between adjacent fronts may have been changed. When smoothing nodes at the front. To ensure that this does not occur. Nm. Edge Ek may have been selected from the existing triangles as in Figure 4. or from a swap operation as in
. the opposite node. Splitting of side edge to maintain even loop
3. The node location can be adjusted incrementally on a vector from the old location to the new location until all neighboring elements are no longer inverted.6 Closing the Front When defining a new side edge. it is possible that the triangles immediately ahead of the front become inverted. Fronts are redefined so that edges now adjacent to both a triangle and a quadrilateral are placed on one of the four state lists and edges no longer adjacent to a triangle are removed from the state lists.5. As tr increases.N k −1 − N j −1 + N j −1 − N j + N k + 1 − N j + 1 + N j +1 − N j + lD = 4+n
∑N
i =1
n
k
− N ti
[4]
The method used for computing lD is decided purely on heuristics.5. may lie on an opposing front. While the Q-Morph algorithm does not require triangles to be near equilateral. as a result of improving the quadrilaterals. it becomes necessary to update the current list of fronts.
Nm Loop 2 Loop 1 Ek EF1 Nk EF2 EF1
Nm Nn Ek Nk EF2
Figure 9. In the case of an inverted element. Selection of side edge forming of new front loop
Figure 10. it is necessary to introduce irregular nodes so that a smooth transition may be afforded. it does rely on the fact that all triangles are uninverted throughout the meshing process. This method is preferred since it tends to produce the fewest number of irregular nodes. then the smoothing method proposed by Blacker (1991) is used unmodified. if the ratio of largest to smallest edge length. on the boundary is less than 2. In addition. By smoothing nodes on the front. and the average length of adjacent edges to Nk is used when tr is greater than 20. as shown in Figure 9. 3.

7 Seams When the angle. To avoid this occurrence. This permits a subsequent side selection operation to define an even number of front edges on adjacent loops. between two adjacent edges on the front is small. If the side edge is to be created from a swap or split operation. as in Figure 5. A front loop may be defined as all edges on a front comprising a continuous unbroken ring. In order to ensure an all-quadrilateral mesh.1991) incorporates a seaming operation. α. Let Nk be the node on the front whose angle. in order to account for triangles ahead of the front. This increases the chances of selecting an existing edge and closing the front. α. Selecting an edge to be used as a new side edge may result in the formation of a loop with an odd number of front edges. nodes Nk-1 and Nk+1. if not already part of the initial triangle mesh. Nn. as shown in Figure 10. is first recovered using Algorithm 1 above. Seaming operation
To accomplish the seam. it is advantageous.Figure 5. Ek can only be used if the number of edges on each resulting front loop is even. the potential number of front edges on the new loop about to be formed is first determined. Ek. The temporary edge. must be merged. In addition to angle α. then a seaming operation is performed. nQ. is split creating a new node. to allow for a larger value of ε. then the selection is made and a new loop is defined. no connection is made. shown in Figure 11. it is required that any individual loop be comprised of an even number of edges. the criteria for seaming is also based on the number of quadrilateral elements. Eo. If the number of edges is even. 3. For this reason. adjacent to the node to be seamed. the edge Eo should first be checked to see if it is part of the opposing front. In either case. its
. when Nm is on an opposing front. satisfies equation 5. Blacker proposes the following criteria for seaming:
α < ε1 for nQ ≥ 5    where ε1 < ε2 α < ε2 otherwise
[5]
α
Nk-1 Eo Nt Nk Nk+1 Nk+1 Nt
Nk
(a) Front to be seamed
(b) Seam closed
Figure 11. Knowing Eo. the operation should not be performed. If the number of edges on the new loop is odd. connecting Nk-1 and Nk+1. Since swapping or splitting Eo would destroy the continuity of the front. Although paving (Blacker. Instead the edge. Any number of loops may be active at a given time in the process of meshing. it must also be defined within the context of the Q-Morph algorithm.

In Figure 12(a). If the ratio of lengths between EF1 and EF2 is greater than 2. edge EF can be processed as a front in state 1-1.Nt. Edges EF. With this information. The longer of the two edges. is first split at its midpoint adding node Nk-1/2 or Nk+1/2. dividing its adjacent triangle and quadrilateral as shown.5. In Figure 12(b).adjacent triangle comprising nodes Nk-1. Transition seam operation
. an optimization based smoothing algorithm (Canann. EF1 is split. the new location of node Nk+1 may result in one or more inverted elements. EFL. In this case. In rare cases. Finally. followed by an update of the states of any adjacent fronts that may have changed. Nk+1. Nt can be deleted. EF1 and EF2 are the edges on the front adjacent to Nk. Another operation described by Blacker (1991) is the transition seam.
Nk-1 EF1 Nk EF2
EF
EFL Nk-1/2
Nk-1
Nk Nk+1
EFR
Nk+1
(a) Split larger of EF1 and EF2
(b)Define EF as front in state 1-1
Nk-1 Nk-1 Nk-1/2 Nk Nk+1 Nk Nk+1 Nk-1/2
(c) Form new transition quad
(d) Smooth and reclassify fronts
Figure 12.Nk. and EFR can then be defined as front edges. requiring only the recovery of the top edge between Nk-1/2 and Nk+1 as in Figure 12(c).1998) is employed which adjusts the node location with the objective of improving a local shape metric for neighboring elements. then a transition seam operation is performed. Local smoothing is then performed.Nk+1. EF1 and EF2. This is required when there is a large difference in size between adjacent fronts. the transition seam is completed after local smoothing and updating of the front as shown in Figure 12(d). all triangles and nodes bounded by the quadrilateral composed of nodes Nk-1. Finally nodes Nk-1 and Nk+1 can be merged at a location midway between their initial positions. With this new configuration. can be determined.

Figure 13(c) shows the configuration after smoothing and reclassification of fronts.1994. which causes its adjacent triangle to be split. The transition split operation is performed when the ratio of lengths between EF1 and EF2 is greater than 2. EFL and EFR. EF can now be defined as a front in state 1-1 and processed to create a new quadrilateral. Each node is moved to the centroid of its neighbors only if an improvement in element shape metric (Lee. the mesh contours can more closely follow the contours of the boundary. Similar to the transition seam.1997. these single operation modifications can be combined into multi-step modifications to further decrease the number of irregular nodes. can be defined as shown in Figure 13(b).1998) operation may be performed.1994) would result. Topological cleanup modifies the connectivity of the quadrilaterals through a series of single-step operations including edge swaps. be introduced as a result of non-orthogonal boundaries or from element size transitions.9 Topological Cleanup and Smoothing Once all of the front edges have been processed and an all-quadrilateral mesh is generated. they may. Irregular nodes may also be introduced when the local nodal density and connectivity provided by the initial triangle mesh is insufficient to generate equilateral quadrilaterals. Let Q1 be the quadrilateral adjacent to the longer of EF1 or EF2. The final smoothing step involves a limited number of iterations of a constrained Laplacian smoothing algorithm. Many of these irregular nodes can be eliminated through local topological cleanup. new front edges. Although similar to the transition seam operation shown in Figure 12. an optimization based smoothing (Canann.5. face opens. Transition split operation
3. Q1 is split into two quadrilaterals and a single triangle. By reducing irregular nodes through topological cleanup. Canann. EF. it is often beneficial to perform local topological cleanup operations to decrease the number of irregular nodes. An additional node is then inserted at the centroid of Q1.
. In situations where Laplacian smoothing produces poor results. Front EF1 in Figure 13(a) is split at its midpoint. and two-edge node removals/insertions (Staten.
Nk-1 Q1 EF1 Nk EF2 Nk+1 Nk EFR Nk+1 Nk+1 Nk-1 EF EFL Nk-1 Nk
(a) Split Q1
(b) Define EF as front in state 1-1
(c) Form transition quad and smooth
Figure 13. As a result. face closes.3. Kinney. as a necessity.8 Transition Split An operation useful for improving transitions is shown in Figure 13. it is applicable when α > ε1 or α > ε2 (see equation 5). In addition. as shown in Figure 13(a). Although Q-Morph attempts to minimize the number of irregular nodes.1997).

The method used for triangulation is unimportant. The toroidal surface of Figure 15 is composed of four surface patches represented as rational B-Splines. The mesh is completed in Figure 14(h) after a final pass of cleanup and smoothing. The first example. Despite using an advancing front scheme. has difficulty maintaining well-aligned rows of elements introducing many irregular internal nodes. Lee’s algorithm shown in Figure 15(b). coupled with an advancing front scheme to combine triangles into quadrilaterals. demonstrates the progression of the Q-Morph algorithm on a simple planar domain with two holes. In this case an advancing front triangle mesher (Canann. Progression of Q-Morph
Figure 15 and Figure 16 compares Q-Morph against Lee’s (1994) quad meshing algorithm.4.1997) was used to create the triangles. Example Problems Four example problems shown in Figure 14 to Figure 18 demonstrate various features of the Q-Morph algorithm.
(a) Initial
(b) 1 Layer
(c) 2 Layers
(d) 3 Layers
(e) 4 Layers
(f) 5 Layers
(g) 6 Layers
(h) Cleanup and smooth
Figure 14. QMorph utilizes projection and geometric evaluation routines as part of the local and final smoothing procedures to maintain nodal locations on the three-dimensional surface. Figure 14(a) shows the initial triangle mesh before Q-Morph begins. provision is made in Q-Morph to mesh loops with smaller elements before those with larger elements. inasmuch as the appropriate nodal density is provided. Figure 16 further illustrates the
. Both Figure 15(a) and (b) were generated using the same initial triangle mesh as well as the same cleanup and smoothing procedures. To improve element transitions. Figure 14(c) shows an additional layer of small elements meshed on the internal circle loop before meshing the larger elements of the outer loop. which uses an indirect method. Figure 14(b)-(g) show the progression of the algorithm as each successive layer of elements is completed. shown in Figure 14.

the Q-Morph algorithm is used as part of a set of meshing tools which also include mapping methods. isotropic quadrilaterals. In order to maintain a specified nodal density near the top of the area. For this example. a sizing function (Owen.
. Surfaces are once again represented by rational B-splines. In this example.Figure 17 demonstrates the use of Q-Morph with a planar surface requiring a high degree of transition.1997) was used during the triangle meshing process.
(a) Partially completed quad mesh
(b) Mesh after cleanup and smoothing
Figure 17. generally towards the interior of the area. Q-Morph is better suited to generating near-equilateral. Large transition mesh for CFD application
The final example in Figure 18 is an industrial application of the Q-Morph algorithm. which can more appropriately create elements of high aspect ratio. Q-Morph forms a single triangle in the mesh. Figure 17(a) shows the partially completed quad mesh with two layers of quads placed. Figure 17(b) shows the same area after final cleanup and smoothing. In practice. each area is first meshed with triangles and then transformed into quadrilaterals. After assigning line divisions. Note that where an odd number of divisions is assigned to an area. The algorithm’s ability to maintain the desired mesh density while still enforcing well-aligned rows of elements transitioning quickly to larger size elements is demonstrated in this example. the model consisting of 104 separate areas was first constructed using a commercial CAD software application. the narrow fillet regions are better represented with a mapped meshing technique. Selection of the appropriate quad meshing method can be done automatically based on the number of lines comprising the area and its aspect ratio.

Times range from 141 to 242 quads converted per CPU second. where times ranged from 313 to 369 quads converted per CPU second. Clean-up and smoothing times were however slower for Figure 17 than for Figure 15 as the transition in element size defined by the quad conversion required additional iterations to converge. 5. various element densities were specified and their results noted. Performance Both speed and element quality of the resulting elements from Q-Morph was evaluated as part of this study. A wide variety of factors can affect the overall speed of the algorithm. For the models in Figure 15 and Figure 17. Tests were performed on a 195 MHz SGI UNIX workstation.Figure 18. Industrial application of Q-Morph combined with mapped meshing
5. For the toroidal surface in Figure 15. Table 1 illustrates two cases where geometry and element transition is critical. times are necessarily affected by the number of geometric evaluations required. This is in contrast to the flat surface of Figure 17.1 Speed Table 1 shows CPU times for both the quad-conversion and the clean-up and smoothing portions of the QMorph algorithm. Table 1 shows performance results from two of the example problems above.
.

0 represents a perfect square. NT and UNIX environments.36 15. Average metrics are however very high. defined by any of the four possible triangles formed by the vertices of the quadrilateral. the Q-Morph algorithm is most beneficial on surfaces where the geometric feature sizes are larger than the specified element size. Metric 0.811 0.2 Element Quality Element quality was measured by shape metric.1). 5.740 0. Table 1 also shows cases where a single triangle is created in the mesh. Q-Morph will be successful. high quality quadrilaterals can be expected.802 0.4
Clean-up and Smoothing Num.370 0.0 represents a quadrilateral with a single corner angle of π.940 0.1998).515 0.170 -0. A β value of 1.31 25. however element quality may suffer.359 0. while a value of 0. Both minimum and average metrics immediately following quad conversion and after clean-up and smoothing are shown in Table 1.19 12.925 0. Metric 0.905 0. As such.2 2. Metric 0.
Triangle to Quad Conversion Model Num.529 0. In some cases. including Windows.893 0.371 0.391 0.44 1. In most cases where these conditions are not met.155 Avg.905 0.382 Avg. Quads 351 1208 4870 19209 Figure 17 727 1892 4472 10581 Num. inverted or poorly shaped quadrilaterals can be created during the quad conversion as indicated by the negative or zero metrics.949 0. α > 0. clean-up and smoothing improved the poorly shaped quads to well within usable limits.155 0.00 -0.889 CPU Time (s) 0.4 136 2. Tris 0 0 0 0 1 0 1 0 Min. Quads 350 1206 4845 19070 696 1785 4288 10231 Num. α. Metric 0.5. For this implementation. In addition.859 0. This occurs automatically in order to resolve situations where an odd number of boundary intervals are specified. Tris 0 0 0 0 1 0 1 0 Min.376 0.9 34.790 0.00 0. it has been successfully ported to a wide variety of platforms. provided the background triangle mesh captures the details of the surface and the background triangles are of reasonable quality (ie.00 0. In all cases tested.817 CPU Time (s) 1.1 33.32 5.45 5.89 10. β similar to that described by Lo (1989) and Canann (1998).936 0. Performance results from Q-Morph
. β is defined as the minimum triangle shape metric.23 4.7 31.344 0.4
Figure 15
Table 1.3 Robustness A diversity of surfaces has been meshed using the Q-Morph algorithm and is currently part of a commercial FEA software release (Ansys. Concave or inverted quadrilaterals may be represented by negative values of β.889 0.948 0.255 0. In general.